Unified relational-theoretic approach in metric-like spaces with an application

In this paper, we introduce a modified implicit relation and obtain some new fixed point results for sigma-implicit type contractive conditions in relational metric-like spaces. We present some nontrivial examples to illustrative facts and compare our results with the related work. We also discuss sufficient conditions for the existence of a unique positive definite solution of the non-linear matrix equation U = D + Sigma(m)(i=1) A(i)*G(U)A(i), where D is an n x n Hermitian positive definite matrix, A(1), A(2), ... , A(m) are n x n matrices, and G is a non-linear self-mapping of the set of all Hermitian matrices which is continuous in the trace norm. Finally, we discuss a couple of examples, convergence and error analysis, average CPU time analysis and visualization of solution in surface plot.